Conductor-discriminant formula
In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Hasse (1926, 1930) for abelian extensions and by Artin (1931) for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension of local or global fields from the Artin conductors of the irreducible characters of the Galois group .
Statement
Let be a finite Galois extension of global fields with Galois group . Then the discriminant equals
where equals the global Artin conductor of .[1]
Example
Let be a cyclotomic extension of the rationals. The Galois group equals . Because is the only finite prime ramified, the global Artin conductor equals the local one . Because is abelian, every non-trivial irreducible character is of degree . Then, the local Artin conductor of equals the conductor of the -adic completion of , i.e. , where is the smallest natural number such that . If , the Galois group is cyclic of order , and by local class field theory and using that one sees easily that : the exponent is
Notes
- Neukirch 1999, VII.11.9.
References
- Artin, Emil (1931), "Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper.", Journal für die Reine und Angewandte Mathematik (in German), 164: 1–11, doi:10.1515/crll.1931.164.1, ISSN 0075-4102, Zbl 0001.00801
- Hasse, H. (1926), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 35: 1–55
- Hasse, H. (1930), "Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper.", Journal für die reine und angewandte Mathematik (in German), 162: 169–184, doi:10.1515/crll.1930.162.169, ISSN 0075-4102
- Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.